Internal problem ID [9858]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.8-1. Equations containing arbitrary
functions (but not containing their derivatives).
Problem number: 8.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {x y^{\prime }-x^{2 n} f \relax (x ) y^{2}-\left (a \,x^{n} f \relax (x )-n \right ) y-f \relax (x ) b=0} \end {gather*}
✓ Solution by Maple
Time used: 0.022 (sec). Leaf size: 64
dsolve(x*diff(y(x),x)=x^(2*n)*f(x)*y(x)^2+(a*x^n*f(x)-n)*y(x)+b*f(x),y(x), singsol=all)
\[ y \relax (x ) = -\frac {\left (\tanh \left (\frac {\sqrt {a^{4}-4 a^{2} b}\, \left (a \left (\int \frac {f \relax (x ) x^{n}}{x}d x \right )+c_{1}\right )}{2 a^{2}}\right ) \sqrt {a^{4}-4 a^{2} b}+a^{2}\right ) x^{-n}}{2 a} \]
✓ Solution by Mathematica
Time used: 5.233 (sec). Leaf size: 109
DSolve[x*y'[x]==x^(2*n)*f[x]*y[x]^2+(a*x^n*f[x]-n)*y[x]+b*f[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt {\frac {a^2}{b}}+\frac {\sqrt {4 b-a^2} \tan \left (\frac {\sqrt {4 b-a^2} \left (\int _1^x\frac {b f(K[5]) \sqrt {\frac {K[5]^{2 n}}{b}}}{K[5]}dK[5]+c_1\right )}{2 \sqrt {b}}\right )}{\sqrt {b}}}{2 \sqrt {\frac {x^{2 n}}{b}}} \\ \end{align*}